Yes. Renter's insurance protects your assets while in the apartment as well as while you are visiting others or if you are forced by disaster to stay away from home.
Renter's insurance is usually inexpensive. If you would not be able to easily replace your possessions if something happened to them, then it is usually worth having.
Yes, you should get insurance to cover your belongings and liability coverage as well. The liability will protect you if you are negligent and someone is injured.
Yes, if there is a fire then the contents of your apartment would be covered if you had renters insurance. Landlord is only responsible for the building not contents.
Yes. When you purchase renters' insurance, you aren't insuring your building, that's the landlord's obligation. Renters' insurance provides coverage for your personal contents and belongings in the event of a peril. Consider purchasing as it is a nominal expense of about $200 per year, depending on deductible.